On Similarity Structure Groups and their W$^*$ and C$^*$-Algebras

TL;DR

This paper introduces CSS$^*$ groups, a subclass of Countable Similarity Structure groups, and explores their algebraic properties, including primeness, non-inner amenability, and $C^*$-simplicity, extending known results and providing new examples.

Contribution

The paper defines CSS$^*$ groups and establishes their key properties, generalizing previous results on Thompson-like groups and analyzing their algebraic and operator algebraic structures.

Findings

CSS$^*$ groups are non-inner amenable and many are properly proximal.

A dichotomy exists: CSS$^*$ groups are either $C^*$-simple with simple commutator subgroup or lack both properties.

CSS$^*$ groups are not acylindrically hyperbolic.

Abstract

Finite Similarity Structure (FSS) groups are a class of generalized Thompson groups first introduced by Farley and Hughes. In this paper, we study the properties of a new subclass of the more general Countable Similarity Structure (CSS) groups, which we call CSS$^*$ groups. The Higman-Thompson groups $V_{d,r}$, the countable R\"over-Nekrashevych groups $V_d(G)$, and the topological full groups of irreducible one-sided subshifts of finite type studied by Matui are all examples of CSS$^*$ groups. One overarching theme is to isolate a class of CSS groups with the necessary properties to generalize the main result of arXiv:2312.08345 -- primeness of the group von Neumann algebra of $V_d$. We achieve this aim for CSS$^*$ groups and in the process prove that CSS$^*$ groups are non-inner amenable and many are properly proximal, which are new results for $V_{d,r}$, $V_d(G)$, and the groups studied by Matui. A second theme is to produce a dichotomy within CSS$^*$ groups; those that are $C^*$-simple and possess a simple commutator subgroup, and those lacking both properties. In particular, we extend the $C^*$-simplicity results of $V$ and $V(G)$ of arXiv:1605.01651, recover the simple commutator subgroup results of arXiv:2008.04791 and arXiv:1210.5800, and give examples that lack both properties. Lastly, we observe that CSS$^*$ groups are not acylindrically hyperbolic, motivating the need to prove many of these results by other methods.